Radius of convergence: $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{100^n}$

Question

I'm having some trouble understanding why the following power series interval of convergences is equal to 0.

$$\sum\limits_{n=0}^{\infty} \frac{n!x^n}{100^n}$$

According to my calculation, my answer is equal to $-100 < x < 100$ since I end up with: $|x/100| < 1$

I did it with the ratio test.

Can somebody explain to me why it equal to $0$? Thank you

Answer 1

Let $a_n:=\frac{n!x^n}{100^n}$. Then for all $x\neq 0$ we have $$\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)|x|}{100}\xrightarrow{n\to\infty}\infty.$$ Factorials beat powers!